Quite surprisingly, for monotone classes of finite graphs the notions
of nowhere denseness, stability, and NIP coincide [1]}. For hereditary classes the picture is
much less clear and stability and NIP do not collapse, and both vastly
generalize the notion of nowhere denseness.
Neither stability nor NIP are hereditary properties, as witnessed by the stable graph class of 1-subdivided cliques.
Its hereditary closure is the class of all 1-subdivided graphs, which is not NIP and (under standard complexity assumptions) does not admit tractable first-order model-checking.
Therefore in the context of hereditary classes, recently the notions of monadically stable and
monadically NIP classes emerged as new possible dividing lines
for tractability (even though, they have long been studied in
classical model theory, see e.g. [2]}).
A class \(\mathrm {{C}}\)  is
monadically stable (resp. monadically NIP) if every monadic expansion of \(\mathrm {{C}}^+\)  (that is, any class
of structures obtained from \(\mathrm {{C}}\)  by extending them by arbitrary unary predicates), is stable (resp. NIP).
Since unary predicates can simulate element removal,
monadic stability/NIP is preserved when taking the hereditary closure of a class.
It was announced by Braunfeld and Laskowski (private communication) that a hereditary class is stable/NIP if and only if it is monadically stable/NIP.
